Abstract
We study conditions under which the commutator subgroup of a finite group G must be “large” in comparison with G.
Introduction
All groups in this paper are finite. For a group G, we denote the center of G and the Frattini subgroup of G by Z(G) and Φ(G), respectively.
In this paper we survey recent results of the authors treating the following general problem:
Problem 1.1Study conditions under which the commutator subgroup G′ of a group G must be “large” in comparison with G.
The authors proved the following result in:
Theorem 1.1Let G ≠ 1 be a group such that Φ(G) = Z(G) = 1. Then |G′| > |G|½.
As will be shown in Section 2, both conditions Φ(G) = 1 and Z(G) = 1 are necessary for the result of Theorem 1.1. Furthermore, as the examples of Frobenius groups of order q(q − 1) (q a power of a prime) show, for any ∈ > 0 there exists a group G ≠ 1 with Φ(G) = Z(G) = 1, such that |G′| ≤ |G|½+∈. Thus the constant ½ can not be replaced by a larger constant in Theorem 1.1. We remark further that the main result in is more general. Indeed, we prove there that if G is a non-abelian group satisfying Φ(G) = 1, then |G′| > [G: Z(G)]1/2.